Abstract algebra

(4) Let H be a subgroup of a group G and assume there are precisely two distinct left cosets of H in G. Prove that these two left cosets of H in G must be H and G \\ H. (Here G \\ H = {g E G I g 9.5 H}.) (5) Prove that if H and K are nite subgroups of a group G and that gcd(#H, #K) = 1, then H 0K = {e}. (6) Prove D2\" (the gorup of symmetries of a regular n-gon) has an element of order 4 if and only if n is even